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### 23.21 Resampling from 2D histograms

As in the one-dimensional case, a two-dimensional histogram made by counting events can be regarded as a measurement of a probability distribution. Allowing for statistical error, the height of each bin represents the probability of an event where (x,y) falls in the range of that bin. For a two-dimensional histogram the probability distribution takes the form p(x,y) dx dy where,

```p(x,y) = n_{ij}/ (N A_{ij})
```

In this equation n_{ij} is the number of events in the bin which contains (x,y), A_{ij} is the area of the bin and N is the total number of events. The distribution of events within each bin is assumed to be uniform.

Data Type: gsl_histogram2d_pdf
`size_t nx, ny`

This is the number of histogram bins used to approximate the probability distribution function in the x and y directions.

`double * xrange`

The ranges of the bins in the x-direction are stored in an array of nx + 1 elements pointed to by xrange.

`double * yrange`

The ranges of the bins in the y-direction are stored in an array of ny + 1 pointed to by yrange.

`double * sum`

The cumulative probability for the bins is stored in an array of nx*ny elements pointed to by sum.

The following functions allow you to create a `gsl_histogram2d_pdf` struct which represents a two dimensional probability distribution and generate random samples from it.

Function: gsl_histogram2d_pdf * gsl_histogram2d_pdf_alloc (size_t nx, size_t ny)

This function allocates memory for a two-dimensional probability distribution of size nx-by-ny and returns a pointer to a newly initialized `gsl_histogram2d_pdf` struct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code of `GSL_ENOMEM`.

Function: int gsl_histogram2d_pdf_init (gsl_histogram2d_pdf * p, const gsl_histogram2d * h)

This function initializes the two-dimensional probability distribution calculated p from the histogram h. If any of the bins of h are negative then the error handler is invoked with an error code of `GSL_EDOM` because a probability distribution cannot contain negative values.

Function: void gsl_histogram2d_pdf_free (gsl_histogram2d_pdf * p)

This function frees the two-dimensional probability distribution function p and all of the memory associated with it.

Function: int gsl_histogram2d_pdf_sample (const gsl_histogram2d_pdf * p, double r1, double r2, double * x, double * y)

This function uses two uniform random numbers between zero and one, r1 and r2, to compute a single random sample from the two-dimensional probability distribution p.

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